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AYATSK

AYATSK is a classification-focused TSK model from the 2025 paper by Xue, Yang, and Wang.

Reference

G. Xue, Y. Yang and J. Wang, "Adaptive Yager T-Norm-Based Takagi–Sugeno–Kang Fuzzy Systems," in IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 55, no. 12, pp. 9802-9815, Dec. 2025, doi: 10.1109/TSMC.2025.3621346.

Mathematical Formulation

Antecedent

The paper uses Composite Exponential membership functions (CEMF) for each input feature:

\[ \mu_{r,d}(x_d) = K^{-1 + \exp\left(-\frac{(x_d - m_{r,d})^2}{2\sigma_{r,d}^2}\right)} \]

The lower bound is \(1/K > 0\), which is important because AYATSK derives the adaptive Yager index from a fixed lower bound rather than from a sample-dependent minimum.

Adaptive Yager T-norm

Rule firing strengths are computed with a parameterized Yager T-norm:

\[ w_r = 1 - \left[\sum_{d=1}^{D} (1 - \mu_{r,d}(x_d))^{\lambda}\right]^{1/\lambda} \]

In the paper, \(\lambda\) is computed once from the input dimensionality \(D\) and the membership lower bound \(\varepsilon\):

\[ \lambda = -\frac{\ln D}{\ln(1 - \varepsilon)} \]

highFIS follows that policy by deriving \(\lambda\) from the number of input features and the CEMF lower bound in AYATSK models.

Defuzzification

AYATSK uses a sum-based defuzzifier to normalize rule activations:

\[ \bar{f}_r = \frac{w_r}{\sum_{i=1}^{R} w_i} \]

This is implemented in highFIS by SumBasedDefuzzifier.

Consequent (first-order)

For classification:

\[ \mathbf{y}_r = W_r \mathbf{x} + \mathbf{b}_r \]

For regression:

\[ \hat{y}_r = \mathbf{w}_r^\top \mathbf{x} + b_r \]

Output aggregation

Final outputs are weighted averages of rule consequents:

  • Classification:
\[ \mathbf{y} = \sum_{r=1}^{R} \bar{f}_r \, \mathbf{y}_r \]
  • Regression:
\[ \hat{y} = \sum_{r=1}^{R} \bar{f}_r \, \hat{y}_r \]

Code ↔ Paper Correspondence

Concept Class / Method Notes
Adaptive Yager T-norm AYATSKClassifierModel, AYATSKRegressorModel Default AYATSK path computes lambda from dimension and lower bound
Yager exponent \(\lambda\) YagerTNorm Fixed after dataset-level adaptation; not learnable per rule
Sum-based defuzzification SumBasedDefuzzifier Normalizes \(\bar{f}_r\) across rules
Composite Exponential MF CompositeExponentialMF Provides a positive lower bound needed by the adaptive strategy
Estimator wrapper AYATSKClassifier, AYATSKRegressor Builds input MFs and handles training/hyperparameters

Implementation notes

  • AYATSKClassifierModel uses MSE loss, Adam, and zero-initialized consequents by default.
  • AYATSKClassifier defaults to n_mfs=3, mf_init="grid", rule_base="coco", epochs=200, and learning_rate=0.001.
  • AYATSKClassifier and AYATSKRegressor expose k (CEMF parameter), with required constraint k > 1.
  • When batch_size=None, the AYATSK estimator uses full-batch for N < 500 and 0.1 * N otherwise, matching the paper’s training policy.
  • AYATSKRegressor remains available as a framework extension, but the paper itself evaluates classification only.

Estimator wrappers

  • AYATSKClassifier wraps AYATSKClassifierModel and supports classification with paper-style adaptive Yager aggregation.
  • AYATSKRegressor wraps AYATSKRegressorModel for regression tasks, but this is outside the paper’s evaluated scope.
  • Estimators accept the usual hyperparameters: n_mfs, mf_init, sigma_scale, random_state, epochs, learning_rate, batch_size, shuffle, validation_data, and patience.
  • pfrb_max_rules exists on the shared estimator base but is unused by AYATSK.

Alignment with the paper

  • The paper evaluates AYATSK on classification datasets only.
  • highFIS implements the paper-strict classification path with CompositeExponentialMF, dataset-level adaptive YagerTNorm, MSE loss, Adam, and zero-initialized consequents.
  • The regressor remains supported as a framework extension, but it should not be presented as part of the paper’s experimental validation.